### Abstract

Let u_{t} + f(u)_{x} = 0 be a strictly hyperbolic n x n system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional Φ = Φ(u, v), equivalent to the L^{1} distance, which is "almost decreasing" i.e., Φ(u(t), v(t)) - Φ(u(s), v(s)) ≦ O (ε) · (t - s) for all t > s ≧ 0, for every pair of ε-approximate solutions u, v with small total variation, generated by a wave front tracking algorithm. The small parameter ε here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in u and in v. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the L^{1} norm. This provides a new proof of the existence of the standard Riemann semigroup generated by a n x n system of conservation laws.

Original language | English (US) |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 149 |

Issue number | 1 |

DOIs | |

State | Published - Oct 22 1999 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

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## Cite this

^{1}stability estimates for n x n conservation laws.

*Archive for Rational Mechanics and Analysis*,

*149*(1), 1-22. https://doi.org/10.1007/s002050050165